Theorem nnssn0s 28290

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28285	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4130	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 4015	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3945   ⊆ wss 3948  {csn 4625   0_s c0s 27848  ℕ_0scnn0s 28282  ℕ_scnns 28283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3466  df-dif 3951  df-ss 3965  df-nns 28285

This theorem is referenced by:  nnssno  28291  nnn0s  28296  nnn0sd  28297  nnsgt0  28306